1. **State the problem:** We need to find the value of $x$ using geometric means from the given right triangles with sides labeled $x$, $x+3$, and segments of length $x$ and $12-x$. 2. **Identify the geometric mean relationship:** In right triangles, the altitude to the hypotenuse is the geometric mean of the two segments it divides the hypotenuse into. Here, the altitude is $x$, and the hypotenuse is divided into segments $x$ and $12 - x$. 3. **Write the formula for the geometric mean:** $$x = \sqrt{x(12 - x)}$$ 4. **Square both sides to eliminate the square root:** $$x^2 = x(12 - x)$$ 5. **Expand and simplify:** $$x^2 = 12x - x^2$$ 6. **Bring all terms to one side:** $$x^2 + x^2 - 12x = 0$$ 7. **Combine like terms:** $$2x^2 - 12x = 0$$ 8. **Factor out the common term:** $$2x(x - 6) = 0$$ 9. **Set each factor equal to zero:** $$2x = 0 \Rightarrow x = 0$$ $$x - 6 = 0 \Rightarrow x = 6$$ 10. **Interpret the solutions:** $x=0$ is not valid for a length, so the solution is: $$\boxed{6}$$ Thus, the value of $x$ is 6.